Need a mathematical equation that produces these types of curves
Weibull to the rescue
There are actually a dozen or more equations that fit this type of data. Some are listed here
We will use the Weibull model
\[
y(t) = \alpha - \beta \cdot e^{-(k \cdot t)^\delta}
\] This models height (y) at time \(t\) as a function of parameters \(\alpha\), \(\beta\), \(k\), and \(\delta\)